NOTE ON (m, q)-ISOMETRIES ON AN HYPERSPACE OF A NORMED SPACE

Given a normed space X we consider the hyperspace k(X) of all non-empty compact convex subsets of X endowed with the Hausdorff distance. We prove that if T : X -> X is an (m, q)-isometry, then it is possible that the map k(T) : k(X) -> k(X), k(T)C := TC, is not an (m, q)-isometry. Moreover, if <(k(X))over cap> is the Radstrom space associated to the hyperspace k(X), then Gamma: k(X) -> k(X) is an (m, q)-isometry if and only if : (T) over cap : <(k(X))over cap> -> <(k(X))over cap> is an (m, q)-isometry.